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We present a comprehensive workflow to simulate single-phase flow and transport in fractured porous media using the discrete fracture matrix approach. The workflow has three primary parts: (1) a method for conforming mesh generation of and around a three-dimensional fracture network, (2) the discretization of the governing equations using a second-order mimetic finite difference method, and (3) implementation of numerical methods for high-performance computing environments. A method to create a conforming Delaunay tetrahedralization of the volume surrounding the fracture network, where the triangular cells of the fracture mesh are faces in the volume mesh, that addresses pathological cases which commonly arise and degrade mesh quality is also provided. Our open-source subsurface simulator uses a hierarchy of process kernels (one kernel per physical process) that allows for both strong and weak coupling of the fracture and matrix domains. We provide verification tests based on analytic solutions for flow and transport, as well as numerical convergence. We also provide multiple expositions of the method in complex fracture networks. In the first example, we demonstrate that the method is robust by considering two scenarios where the fracture network acts as a barrier to flow, as the primary pathway, or offers the same resistance as the surrounding matrix. In the second test, flow and transport through a three-dimensional stochastically generated network containing 257 fractures is presented.

Stochastic modeling has become a popular approach to quantify uncertainty in flows through heterogeneous porous media. The uncertainty in heterogeneous structure properties is often parameterized by a high-dimensional random variable. This leads to a deterministic problem in a high-dimensional parameter space and the numerical computation becomes very challengeable as the dimension of the parameter space increases. To efficiently tackle the high-dimensionality, we propose a hybrid high dimensional model representation (HDMR) technique, through which the high-dimensional stochastic model is decomposed into a moderate-dimensional stochastic model in a most active random space and a few one-dimensional stochastic models. The derived low-dimensional stochastic models are solved by incorporating sparse grid stochastic collocation method into the proposed hybrid HDMR. The porous media properties such as permeability are often heterogeneous. To treat the heterogeneity, we use a mixed multiscale finite element method (MMsFEM) to simulate each of derived stochastic models. To capture the non-local spatial features of the porous media and the important effects of random variables, we can hierarchically incorporate the global information individually from each of random parameters. This significantly enhances the accuracy of the multiscale simulation. The synergy of the hybrid HDMR and the MMsFEM reduces the stochastic model of flows in both stochastic space and physical space, and significantly decreases the computation complexity. We carefully analyze the proposed HDMR technique and the derived stochastic MMsFEM. A few numerical experiments are carried out for two-phase flows in random porous media and support the efficiency and accuracy of the MMsFEM based on the hybrid HDMR., Comment: 32 pages, 14 figures

We develop a family of expanded mixed Multiscale Finite Element Methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity and Lagrange multipliers. We use multiscale basis functions for the both velocity and gradient of pressure. In the expanded mixed MsFEM framework, we consider both cases of separable-scale and non-separable spatial scales. We specifically analyze the methods in three categories: periodic separable scales, $G$- convergence separable scales, and continuum scales. When there is no scale separation, using some global information can improve accuracy for the expanded mixed MsFEMs. We present rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes both conforming and nonconforming expanded mixed MsFEM. Numerical results are presented for various multiscale models and flows in porous media with shales to illustrate the efficiency of the expanded mixed MsFEMs., Comment: 33 pages